Functional analysis is an abstract branch of mathematics that grew out of classical analysis. It represents one of the most important branches of the mathematical sciences. Together with abstract algebra and mathematical analysis, it serves as a foundation of many other branches of mathematics. Functional analysis is in particular widely used in probability and random function theory, numerical analysis, mathematical physics and their numerous applications. It serves as a powerful tool in modern control and information sciences.

The development of the subject started from the beginning of the twentieth century, mainly through the initiative of the Russian school mathematicians. The impetus came from the developments of linear algebra, linear ordinary and partial differential equations, calculus of variation, approximation theory and, in particular, those of linear integral equations, the theory of which had the greatest impact on the development and promotion of modern ideas. Mathematicians observed that problems from different fields often possess related features and properties. This allowed for an effective unifying approach towards the problems, the unification being obtained by the omission of inessential details. Hence the advantage of such an abstract approach is that it concentrates on the essential facts, so that they become clearly visible.

In 4.2.3 we defined a bounded linear operator in the setting of two normed linear spaces Ex and Ey and studied several interesting properties bounded linear operators. But if said operator ceases to be bounded, then we get an unbounded linear operator.

The class of unbounded linear operators include a rich class of operators, notably the class of differential operators. In 4.2.11 we gave an example of an unbounded differential operator. There are usually two different approaches to treating a differential operator in the usual function space setting. The first is to define a new topology on the space so that the differential operators are continuous on a nonnormable topological linear space. This is known as L. Schwartz's theory of distribution (Schwartz, [52]). The other approach is to retain the Banach space structure while developing and applying the general theory of unbounded linear operators (Browder, F [9]). We will use the second approach. We have already introduced closed operators in Chapter 7. The linear differential operators are usually closed operators, or at least have closed linear extensions. Closed linear operators and continuous linear operators have some common features in that many theorems which hold true for continuous linear operators are also true for closed linear operators. In this chapter we point out some salient features of the class of unbounded linear operators.

This chapter focusses on a natural and useful generalisation of bounded linear operators having a finite dimensional range. The concept of a compact linear operator is introduced in section 8.1. Compact linear operators often appear in applications. They play a crucial role in the theory of integral equations and in various problems of mathematical physics. The relation of compactness with weak convergence and reflexivity is highlighted. The spectral properties of a compact linear operator are studied in section 8.2. The notion of the Fredholm alternative and the relevant theorems are provided in section 8.3. Section 8.4 shows how to construct a finite rank approximations of a compact operator. A reduction of the finite rank problem to a finite dimensional problem is also given.

Compact Linear Operators

Definition: compact linear operator

A linear operator mapping a normed linear space Ex onto a normed linear space Ey is said to be compact if it maps a bounded set of (Ex) into a compact set of (Ey).

In this chapter we recapitulate the mathematical preliminaries that will be relevant to the development of functional analysis in later chapters. This chapter comprises six sections. We presume that the reader has been exposed to an elementary course in real analysis and linear algebra.

Set

The theory of sets is one of the principal tools of mathematics. One type of study of set theory addresses the realm of logic, philosophy and foundations of mathematics. The other study goes into the highlands of mathematics, where set theory is used as a medium of expression for various concepts in mathematics. We assume that the sets are ‘not too big’ to avoid any unnecessary contradiction. In this connection one can recall the famous ‘Russell's Paradox’ (Russell, 1959). A set is a collection of distinct and distinguishable objects. The objects that belong to a set are called elements, members or points of the set. If an object a belongs to a set A, then we write a ∈ A. On the other hand, if a does not belong to A, we write a ∉ A. A set may be described by listing the elements and enclosing them in braces. For example, the set A formed out of the letters a, a, a, b, b, c can be expressed as A = {a, b, c}. A set can also be described by some defining properties. For example, the set of natural numbers can be written as ℕ = {x : x, a natural number} or {x|x, a natural number}.

This book is the outgrowth of the lectures delivered on functional analysis and allied topics to the postgraduate classes in the Department of Applied Mathematics, Calcutta University, India. I feel I owe an explanation as to why I should write a new book, when a large number of books on functional analysis at the elementary level are available. Behind every abstract thought there is a concrete structure. I have tried to unveil the motivation behind every important development of the subject matter. I have endeavoured to make the presentation lucid and simple so that the learner can read without outside help.

The first chapter, entitled ‘Preliminaries’, contains discussions on topics of which knowledge will be necessary for reading the later chapters. The first concepts introduced are those of a set, the cardinal number, the different operations on a set and a partially ordered set respectively. Important notions like Zorn's lemma, Zermelo's axiom of choice are stated next. The concepts of a function and mappings of different types are introduced and exhibited with examples. Next comes the notion of a linear space and examples of different types of linear spaces. The definition of subspace and the notion of linear dependence or independence of members of a subspace are introduced. Ideas of partition of a space as a direct sum of subspaces and quotient space are explained. ‘Metric space’ as an abstraction of real line ℝ is introduced.

An Introduction to Wavelet Analysis

The concept of Wavelet was first introduced around 1980. It came out as a synthesis of ideas borrowed from disciplines including mathematics (Calderáon Zygmund operators and Littlewood-Paley theory), physics (coherent states formalism in quantum mechanism and renormalizing group) and engineering (quadratic mirror filters, sidebend coding in signal processing and pyramidal algorithms in image processing) (Debnath [17]).

Wavelet analysis provides a systematic new way to represent and analyze multiscale structures. The special feature of Wavelet analysis is to generalize and expand the representations of functions by orthogonal basis to infinite domains. For this purpose, compactly supported [see 13.5] basis functions are used and this linear combination represents the function. These are the kinds of functions that are realized by physical devices.

A Hilbert space is a Banach space endowed with a dot product or scalar product. A normed linear space has a norm, or the concept of distance, but does not admit the concept of the angle between two elements or two vectors. But an inner product space admits both the concepts such as the concept of distance or norm and the concept of orthogonality–in other words, the angle between two vectors. Just as a complete normed linear space is called a Banach space, a complete inner product space is called a Hilbert space. An inner product space is a generalisation of the n-dimensional Euclidean space to infinite dimensions.

The whole theory was initiated by the work of D. Hilbert (1912) [24] on integral equations. The currently used geometrical notation and terminology is analogous to that of Euclidean geometry and was coined by E. Schmidt (1908) [50]. These spaces have up to now been the most useful spaces in practical applications of functional analysis.